When \(\operatorname{Map} (\Sigma )\)  is CB generated, there is a finite set
\(F \subset \operatorname{FMap} (\Sigma )\)  such that \(\nu _L \cup F\)  generates \(\operatorname{FMap} \) 
(See [1]} for details).
We say \(\pi \in \mathrm {Sym}(M_{\text{i}so})\)  is type preserving
if for every \(x_A \in M_{\text{i}so}\) , \(x_B=\pi (x_A) \in E(x_A)\) .
For every type preserving \(\pi \in \mathrm {Sym}(M_{\text{i}so})\) 
we fix a homeomorphism \(p_\pi \)  which permutes the sets in \(\mathcal {A}\) ,
sending \(A\)  to \(B\)  when \(\pi (x_A) = x_B\) , but fixes the finite type
subsurface \(L\)  set wise (that is the restriction of \(p_\pi \)  to \(L\)  is
an element of \(\operatorname{Map} (L)\) ). In fact, we can choose these so that
\(p_{\pi ^{-1}} = p_\pi ^{-1}\) .
Since there are only finitely many such permutations, the collection \(P = \lbrace  p_\pi \rbrace \) 
is finite. Then \(\nu _L \cup F \cup P\)  is a CB generating sets for \(\operatorname{Map} (\Sigma )\) .
